A248752 Decimal expansion of limit of the imaginary part of f(1-i,n)/f(1-i,n+1), where f(x,n) is the n-th Fibonacci polynomial.
2, 5, 7, 0, 6, 5, 8, 6, 4, 1, 2, 1, 6, 7, 7, 1, 6, 0, 9, 0, 8, 5, 6, 8, 0, 6, 2, 0, 5, 2, 7, 3, 3, 7, 1, 8, 9, 0, 3, 7, 5, 7, 0, 0, 7, 9, 9, 8, 8, 1, 3, 4, 9, 4, 5, 2, 4, 1, 3, 0, 7, 9, 3, 7, 8, 0, 9, 4, 2, 2, 3, 6, 0, 4, 3, 1, 2, 1, 4, 5, 0, 9, 4, 0, 7, 6
Offset: 0
Examples
limit = 0.2570658641216771609085680620527337189037570... Let q(x,n) = f(x,n)/f(x,n+1) and c = 1-i. n f(x,n) Re(q(c,n)) Im(q(c,n)) 1 1 1/2 1/2 2 x 3/5 1/5 3 1 + x^2 1/2 1/4 4 2*x + x^3 8/15 4/15 5 1 + 3*x^2 + x^4 69/130 33/130 Re(q(1-i,11)) = 5021/9490 = 0.5290832... Im(q(1-i,11)) = 4879/18980 = 0.257060...
Programs
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Maple
evalf((1-sqrt(sqrt(5)-2))/2, 120); # Vaclav Kotesovec, Oct 19 2014
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Mathematica
z = 300; t = Table[Fibonacci[n, x]/Fibonacci[n + 1, x], {n, 1, z}]; u = t /. x -> 1 - I; d1 = N[Re[u][[z]], 130] d2 = N[Im[u][[z]], 130] r1 = RealDigits[d1] (* A248751 *) r2 = RealDigits[d2] (* A248752 *)
Formula
Equals (1-sqrt(sqrt(5)-2))/2. - Vaclav Kotesovec, Oct 19 2014
From Wolfdieter Lang, Mar 02 2018: (Start)
Equals (1 - (2 - phi)*sqrt(phi))/2, with phi = A001622.
Equals (1/10)*y*(1 - (1/50)*y^2) with y = A300070. (End)
Comments